Everything on this page is the growing pentagrid with n = 7. Seven families of parallel lines instead of five, turned 2π/7 apart; every crossing still becomes a tile; every line still stays fastened to the edge midpoints it passes between and gets dragged into a ribbon. Not a line of the drawing code knows which it is running.
What changes is worth watching for. There are three rhombs, not two. A grid of order n makes ⌊n/2⌋ shapes, one for each way two families can be separated, with corner angle 2πc/n. Five gives the familiar pair at 72° and 144°. Seven gives 51.43°, 102.86° and 154.29° — and they do not appear equally often. In the limit each class shows up in proportion to |sin 2πc/n|, the rate at which two families at that angle cross, which works out at 35.7%, 44.5% and 19.8%.
band at 0 leaves bare lines · 100% fills the tile
Drag to pan, scroll to zoom.
The page opens at Σγ = 1, which with the offsets split evenly is every γj = 1/7. That is the one setting with genuine seven-fold symmetry about the origin. Push the total to its far end, 7/2, and every offset is ½ instead: that one is fourteen-fold. Both are Lutfalla's Theorem 1, and both are claims about the offsets being equal — dualization commutes with rotation about the origin, so the tiling inherits whatever symmetry the grid has, and spreading the dials apart leaves it with none. Move a single dial and watch the note stop claiming it.
This is not a Penrose tiling and there is no inflation rule waiting to be found. It is a cut-and-project tiling, quasiperiodic and edge-to-edge, and that is all — de Bruijn's pentagrid theory is about five. Two other things stay behind. The Wieringa roof does not exist here: lifting works at five because ℝ5 splits into the plane, one perpendicular direction and the all-ones line, so a single height z = ½ ∑K does the job. In ℝ7 the perpendicular space is five-dimensional and there is no height function to find. And the exact regularity test stays at five: the pentagrid criterion works because the coefficients live in ℝ(φ), a quadratic field, so one real equation forces two rational ones. At seven the field is cubic and there are 35 triples rather than 10. Lutfalla remarks that de Bruijn's characterisation “is not easily generalised”, and does not try — his Theorem 2 gives a sufficient condition instead, which is what the meter reads here: keep every offset a non-integer rational and an odd-order grid is regular. So the meter can promise you regularity, but it can no longer name the guilty triples.
One thing that does generalise cleanly is the registration gain. The tiling is drawn n/2 times the size of the grid that makes it, because ∑vjvjT = (n/2)I for any n equally spaced unit vectors. The pentagrid's 5/2 becomes 7/2, and the tiles still start life exactly on the crossings that made them.